第 22147 題：有限因數格上的Möbius反演

原題與已知條件

positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={"n":552,"a":22,"b":1,"omitted_divisor":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={"n":552,"a":22,"b":1,"omitted_divisor":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-mobius-inversion-finite-divisors

逐步推導

source operation=mobinv-incomplete-divisor-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary]

Mobius inversion ORIGINAL inverse:actual positive input=552;check complete integer range1..552

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(552, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 552

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=552;check complete integer range1..552

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(22, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor := mul(22, 4)

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 88

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b := sum(88, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 89

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(22, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 132

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(132, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 133

Mobius inversion ORIGINAL inverse F(n/1) f(8) ORIGINAL a times divisor := mul(22, 8)

Mobius inversion ORIGINAL inverse F(n/1) f(8) ORIGINAL a times divisor = 176

Mobius inversion ORIGINAL inverse F(n/1) f(8) plus ORIGINAL b := sum(176, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(8) plus ORIGINAL b = 177

Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor := mul(22, 12)

Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor = 264

Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b := sum(264, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b = 265

Mobius inversion ORIGINAL inverse F(n/1) f(23) ORIGINAL a times divisor := mul(22, 23)

Mobius inversion ORIGINAL inverse F(n/1) f(23) ORIGINAL a times divisor = 506

Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b := sum(506, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b = 507

Mobius inversion ORIGINAL inverse F(n/1) f(24) ORIGINAL a times divisor := mul(22, 24)

Mobius inversion ORIGINAL inverse F(n/1) f(24) ORIGINAL a times divisor = 528

Mobius inversion ORIGINAL inverse F(n/1) f(24) plus ORIGINAL b := sum(528, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(24) plus ORIGINAL b = 529

Mobius inversion ORIGINAL inverse F(n/1) f(46) ORIGINAL a times divisor := mul(22, 46)

Mobius inversion ORIGINAL inverse F(n/1) f(46) ORIGINAL a times divisor = 1012

Mobius inversion ORIGINAL inverse F(n/1) f(46) plus ORIGINAL b := sum(1012, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(46) plus ORIGINAL b = 1013

Mobius inversion ORIGINAL inverse F(n/1) f(69) ORIGINAL a times divisor := mul(22, 69)

Mobius inversion ORIGINAL inverse F(n/1) f(69) ORIGINAL a times divisor = 1518

Mobius inversion ORIGINAL inverse F(n/1) f(69) plus ORIGINAL b := sum(1518, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(69) plus ORIGINAL b = 1519

Mobius inversion ORIGINAL inverse F(n/1) f(92) ORIGINAL a times divisor := mul(22, 92)

Mobius inversion ORIGINAL inverse F(n/1) f(92) ORIGINAL a times divisor = 2024

Mobius inversion ORIGINAL inverse F(n/1) f(92) plus ORIGINAL b := sum(2024, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(92) plus ORIGINAL b = 2025

Mobius inversion ORIGINAL inverse F(n/1) f(138) ORIGINAL a times divisor := mul(22, 138)

Mobius inversion ORIGINAL inverse F(n/1) f(138) ORIGINAL a times divisor = 3036

Mobius inversion ORIGINAL inverse F(n/1) f(138) plus ORIGINAL b := sum(3036, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(138) plus ORIGINAL b = 3037

Mobius inversion ORIGINAL inverse F(n/1) f(184) ORIGINAL a times divisor := mul(22, 184)

Mobius inversion ORIGINAL inverse F(n/1) f(184) ORIGINAL a times divisor = 4048

Mobius inversion ORIGINAL inverse F(n/1) f(184) plus ORIGINAL b := sum(4048, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(184) plus ORIGINAL b = 4049

Mobius inversion ORIGINAL inverse F(n/1) f(276) ORIGINAL a times divisor := mul(22, 276)

Mobius inversion ORIGINAL inverse F(n/1) f(276) ORIGINAL a times divisor = 6072

Mobius inversion ORIGINAL inverse F(n/1) f(276) plus ORIGINAL b := sum(6072, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(276) plus ORIGINAL b = 6073

Mobius inversion ORIGINAL inverse F(n/1) f(552) ORIGINAL a times divisor := mul(22, 552)

Mobius inversion ORIGINAL inverse F(n/1) f(552) ORIGINAL a times divisor = 12144

Mobius inversion ORIGINAL inverse F(n/1) f(552) plus ORIGINAL b := sum(12144, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(552) plus ORIGINAL b = 12145

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(23, 45, 67, 89, 133, 177, 265, 507, 529, 1013, 1519, 2025, 3037, 4049, 6073, 12145)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 31696

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 31696)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 31696

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=2>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(552, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 276

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=276;check complete integer range1..276

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12, 23, 46, 69, 92, 138, 276];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(22, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/2) f(4) ORIGINAL a times divisor := mul(22, 4)

Mobius inversion ORIGINAL inverse F(n/2) f(4) ORIGINAL a times divisor = 88

Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b := sum(88, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b = 89

Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor := mul(22, 6)

Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor = 132

Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b := sum(132, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b = 133

Mobius inversion ORIGINAL inverse F(n/2) f(12) ORIGINAL a times divisor := mul(22, 12)

Mobius inversion ORIGINAL inverse F(n/2) f(12) ORIGINAL a times divisor = 264

Mobius inversion ORIGINAL inverse F(n/2) f(12) plus ORIGINAL b := sum(264, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(12) plus ORIGINAL b = 265

Mobius inversion ORIGINAL inverse F(n/2) f(23) ORIGINAL a times divisor := mul(22, 23)

Mobius inversion ORIGINAL inverse F(n/2) f(23) ORIGINAL a times divisor = 506

Mobius inversion ORIGINAL inverse F(n/2) f(23) plus ORIGINAL b := sum(506, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(23) plus ORIGINAL b = 507

Mobius inversion ORIGINAL inverse F(n/2) f(46) ORIGINAL a times divisor := mul(22, 46)

Mobius inversion ORIGINAL inverse F(n/2) f(46) ORIGINAL a times divisor = 1012

Mobius inversion ORIGINAL inverse F(n/2) f(46) plus ORIGINAL b := sum(1012, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(46) plus ORIGINAL b = 1013

Mobius inversion ORIGINAL inverse F(n/2) f(69) ORIGINAL a times divisor := mul(22, 69)

Mobius inversion ORIGINAL inverse F(n/2) f(69) ORIGINAL a times divisor = 1518

Mobius inversion ORIGINAL inverse F(n/2) f(69) plus ORIGINAL b := sum(1518, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(69) plus ORIGINAL b = 1519

Mobius inversion ORIGINAL inverse F(n/2) f(92) ORIGINAL a times divisor := mul(22, 92)

Mobius inversion ORIGINAL inverse F(n/2) f(92) ORIGINAL a times divisor = 2024

Mobius inversion ORIGINAL inverse F(n/2) f(92) plus ORIGINAL b := sum(2024, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(92) plus ORIGINAL b = 2025

Mobius inversion ORIGINAL inverse F(n/2) f(138) ORIGINAL a times divisor := mul(22, 138)

Mobius inversion ORIGINAL inverse F(n/2) f(138) ORIGINAL a times divisor = 3036

Mobius inversion ORIGINAL inverse F(n/2) f(138) plus ORIGINAL b := sum(3036, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(138) plus ORIGINAL b = 3037

Mobius inversion ORIGINAL inverse F(n/2) f(276) ORIGINAL a times divisor := mul(22, 276)

Mobius inversion ORIGINAL inverse F(n/2) f(276) ORIGINAL a times divisor = 6072

Mobius inversion ORIGINAL inverse F(n/2) f(276) plus ORIGINAL b := sum(6072, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(276) plus ORIGINAL b = 6073

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(23, 45, 67, 89, 133, 265, 507, 1013, 1519, 2025, 3037, 6073)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 14796

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 14796)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -14796

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=3>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3

complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(552, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 184

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=184;check complete integer range1..184

ALL positive divisors INCREASING=[1, 2, 4, 8, 23, 46, 92, 184];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor := mul(22, 4)

Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor = 88

Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b := sum(88, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b = 89

Mobius inversion ORIGINAL inverse F(n/3) f(8) ORIGINAL a times divisor := mul(22, 8)

Mobius inversion ORIGINAL inverse F(n/3) f(8) ORIGINAL a times divisor = 176

Mobius inversion ORIGINAL inverse F(n/3) f(8) plus ORIGINAL b := sum(176, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(8) plus ORIGINAL b = 177

Mobius inversion ORIGINAL inverse F(n/3) f(23) ORIGINAL a times divisor := mul(22, 23)

Mobius inversion ORIGINAL inverse F(n/3) f(23) ORIGINAL a times divisor = 506

Mobius inversion ORIGINAL inverse F(n/3) f(23) plus ORIGINAL b := sum(506, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(23) plus ORIGINAL b = 507

Mobius inversion ORIGINAL inverse F(n/3) f(46) ORIGINAL a times divisor := mul(22, 46)

Mobius inversion ORIGINAL inverse F(n/3) f(46) ORIGINAL a times divisor = 1012

Mobius inversion ORIGINAL inverse F(n/3) f(46) plus ORIGINAL b := sum(1012, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(46) plus ORIGINAL b = 1013

Mobius inversion ORIGINAL inverse F(n/3) f(92) ORIGINAL a times divisor := mul(22, 92)

Mobius inversion ORIGINAL inverse F(n/3) f(92) ORIGINAL a times divisor = 2024

Mobius inversion ORIGINAL inverse F(n/3) f(92) plus ORIGINAL b := sum(2024, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(92) plus ORIGINAL b = 2025

Mobius inversion ORIGINAL inverse F(n/3) f(184) ORIGINAL a times divisor := mul(22, 184)

Mobius inversion ORIGINAL inverse F(n/3) f(184) ORIGINAL a times divisor = 4048

Mobius inversion ORIGINAL inverse F(n/3) f(184) plus ORIGINAL b := sum(4048, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(184) plus ORIGINAL b = 4049

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(23, 45, 89, 177, 507, 1013, 2025, 4049)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 7928

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 7928)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -7928

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=4>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(4, 2)

upf extract divisor2 occurrence0 = 2

upf extract divisor2 occurrence1 := div(2, 2)

upf extract divisor2 occurrence1 = 1

next divisor=2;previous remaining=4;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(552, 4)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 138

Mobius inversion ORIGINAL inverse F(n/4):actual positive input=138;check complete integer range1..138

ALL positive divisors INCREASING=[1, 2, 3, 6, 23, 46, 69, 138];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/4) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/4) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor := mul(22, 3)

Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/4) f(6) ORIGINAL a times divisor := mul(22, 6)

Mobius inversion ORIGINAL inverse F(n/4) f(6) ORIGINAL a times divisor = 132

Mobius inversion ORIGINAL inverse F(n/4) f(6) plus ORIGINAL b := sum(132, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(6) plus ORIGINAL b = 133

Mobius inversion ORIGINAL inverse F(n/4) f(23) ORIGINAL a times divisor := mul(22, 23)

Mobius inversion ORIGINAL inverse F(n/4) f(23) ORIGINAL a times divisor = 506

Mobius inversion ORIGINAL inverse F(n/4) f(23) plus ORIGINAL b := sum(506, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(23) plus ORIGINAL b = 507

Mobius inversion ORIGINAL inverse F(n/4) f(46) ORIGINAL a times divisor := mul(22, 46)

Mobius inversion ORIGINAL inverse F(n/4) f(46) ORIGINAL a times divisor = 1012

Mobius inversion ORIGINAL inverse F(n/4) f(46) plus ORIGINAL b := sum(1012, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(46) plus ORIGINAL b = 1013

Mobius inversion ORIGINAL inverse F(n/4) f(69) ORIGINAL a times divisor := mul(22, 69)

Mobius inversion ORIGINAL inverse F(n/4) f(69) ORIGINAL a times divisor = 1518

Mobius inversion ORIGINAL inverse F(n/4) f(69) plus ORIGINAL b := sum(1518, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(69) plus ORIGINAL b = 1519

Mobius inversion ORIGINAL inverse F(n/4) f(138) ORIGINAL a times divisor := mul(22, 138)

Mobius inversion ORIGINAL inverse F(n/4) f(138) ORIGINAL a times divisor = 3036

Mobius inversion ORIGINAL inverse F(n/4) f(138) plus ORIGINAL b := sum(3036, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(138) plus ORIGINAL b = 3037

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(23, 45, 67, 133, 507, 1013, 1519, 3037)

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 6344

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, 6344)

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=6>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(6, 2)

upf extract divisor2 occurrence0 = 3

next divisor=2;previous remaining=6;all smaller divisors exhausted

extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6

complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]

mu=1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(552, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 92

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=92;check complete integer range1..92

ALL positive divisors INCREASING=[1, 2, 4, 23, 46, 92];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/6) f(4) ORIGINAL a times divisor := mul(22, 4)

Mobius inversion ORIGINAL inverse F(n/6) f(4) ORIGINAL a times divisor = 88

Mobius inversion ORIGINAL inverse F(n/6) f(4) plus ORIGINAL b := sum(88, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(4) plus ORIGINAL b = 89

Mobius inversion ORIGINAL inverse F(n/6) f(23) ORIGINAL a times divisor := mul(22, 23)

Mobius inversion ORIGINAL inverse F(n/6) f(23) ORIGINAL a times divisor = 506

Mobius inversion ORIGINAL inverse F(n/6) f(23) plus ORIGINAL b := sum(506, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(23) plus ORIGINAL b = 507

Mobius inversion ORIGINAL inverse F(n/6) f(46) ORIGINAL a times divisor := mul(22, 46)

Mobius inversion ORIGINAL inverse F(n/6) f(46) ORIGINAL a times divisor = 1012

Mobius inversion ORIGINAL inverse F(n/6) f(46) plus ORIGINAL b := sum(1012, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(46) plus ORIGINAL b = 1013

Mobius inversion ORIGINAL inverse F(n/6) f(92) ORIGINAL a times divisor := mul(22, 92)

Mobius inversion ORIGINAL inverse F(n/6) f(92) ORIGINAL a times divisor = 2024

Mobius inversion ORIGINAL inverse F(n/6) f(92) plus ORIGINAL b := sum(2024, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(92) plus ORIGINAL b = 2025

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(23, 45, 89, 507, 1013, 2025)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 3702

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 3702)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 3702

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor8:actual positive input for mu=8

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=8>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(8, 2)

upf extract divisor2 occurrence0 = 4

upf extract divisor2 occurrence1 := div(4, 2)

upf extract divisor2 occurrence1 = 2

upf extract divisor2 occurrence2 := div(2, 2)

upf extract divisor2 occurrence2 = 1

next divisor=2;previous remaining=8;all smaller divisors exhausted

extracted multiplicity=3;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=8

complete increasing prime multiset=[2, 2, 2];full product=8;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor8:input=8;COMPLETE prime multiset=[2, 2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8 := div(552, 8)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8 = 69

Mobius inversion ORIGINAL inverse F(n/8):actual positive input=69;check complete integer range1..69

ALL positive divisors INCREASING=[1, 3, 23, 69];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/8) f(3) ORIGINAL a times divisor := mul(22, 3)

Mobius inversion ORIGINAL inverse F(n/8) f(3) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/8) f(3) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/8) f(3) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/8) f(23) ORIGINAL a times divisor := mul(22, 23)

Mobius inversion ORIGINAL inverse F(n/8) f(23) ORIGINAL a times divisor = 506

Mobius inversion ORIGINAL inverse F(n/8) f(23) plus ORIGINAL b := sum(506, 1)

Mobius inversion ORIGINAL inverse F(n/8) f(23) plus ORIGINAL b = 507

Mobius inversion ORIGINAL inverse F(n/8) f(69) ORIGINAL a times divisor := mul(22, 69)

Mobius inversion ORIGINAL inverse F(n/8) f(69) ORIGINAL a times divisor = 1518

Mobius inversion ORIGINAL inverse F(n/8) f(69) plus ORIGINAL b := sum(1518, 1)

Mobius inversion ORIGINAL inverse F(n/8) f(69) plus ORIGINAL b = 1519

Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum := sum(23, 67, 507, 1519)

Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum = 2116

Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8) := mul(0, 2116)

Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:actual positive input for mu=12

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=12>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(12, 2)

upf extract divisor2 occurrence0 = 6

upf extract divisor2 occurrence1 := div(6, 2)

upf extract divisor2 occurrence1 = 3

next divisor=2;previous remaining=12;all smaller divisors exhausted

extracted multiplicity=2;next remaining=3;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 3)

upf canonical complete product factor2 = 12

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=12

complete increasing prime multiset=[2, 2, 3];full product=12;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:input=12;COMPLETE prime multiset=[2, 2, 3];distinct primes=[2, 3]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 := div(552, 12)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 = 46

Mobius inversion ORIGINAL inverse F(n/12):actual positive input=46;check complete integer range1..46

ALL positive divisors INCREASING=[1, 2, 23, 46];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/12) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/12) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/12) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/12) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/12) f(23) ORIGINAL a times divisor := mul(22, 23)

Mobius inversion ORIGINAL inverse F(n/12) f(23) ORIGINAL a times divisor = 506

Mobius inversion ORIGINAL inverse F(n/12) f(23) plus ORIGINAL b := sum(506, 1)

Mobius inversion ORIGINAL inverse F(n/12) f(23) plus ORIGINAL b = 507

Mobius inversion ORIGINAL inverse F(n/12) f(46) ORIGINAL a times divisor := mul(22, 46)

Mobius inversion ORIGINAL inverse F(n/12) f(46) ORIGINAL a times divisor = 1012

Mobius inversion ORIGINAL inverse F(n/12) f(46) plus ORIGINAL b := sum(1012, 1)

Mobius inversion ORIGINAL inverse F(n/12) f(46) plus ORIGINAL b = 1013

Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum := sum(23, 45, 507, 1013)

Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum = 1588

Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) := mul(0, 1588)

Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor23:actual positive input for mu=23

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=23>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

next divisor=2;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo3=2;extracted multiset=[]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 23)

upf canonical complete product factor0 = 23

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=23

complete increasing prime multiset=[23];full product=23;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor23:input=23;COMPLETE prime multiset=[23];distinct primes=[23]

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 := div(552, 23)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 = 24

Mobius inversion ORIGINAL inverse F(n/23):actual positive input=24;check complete integer range1..24

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 24];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/23) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/23) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/23) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/23) f(3) ORIGINAL a times divisor := mul(22, 3)

Mobius inversion ORIGINAL inverse F(n/23) f(3) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/23) f(3) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(3) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/23) f(4) ORIGINAL a times divisor := mul(22, 4)

Mobius inversion ORIGINAL inverse F(n/23) f(4) ORIGINAL a times divisor = 88

Mobius inversion ORIGINAL inverse F(n/23) f(4) plus ORIGINAL b := sum(88, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(4) plus ORIGINAL b = 89

Mobius inversion ORIGINAL inverse F(n/23) f(6) ORIGINAL a times divisor := mul(22, 6)

Mobius inversion ORIGINAL inverse F(n/23) f(6) ORIGINAL a times divisor = 132

Mobius inversion ORIGINAL inverse F(n/23) f(6) plus ORIGINAL b := sum(132, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(6) plus ORIGINAL b = 133

Mobius inversion ORIGINAL inverse F(n/23) f(8) ORIGINAL a times divisor := mul(22, 8)

Mobius inversion ORIGINAL inverse F(n/23) f(8) ORIGINAL a times divisor = 176

Mobius inversion ORIGINAL inverse F(n/23) f(8) plus ORIGINAL b := sum(176, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(8) plus ORIGINAL b = 177

Mobius inversion ORIGINAL inverse F(n/23) f(12) ORIGINAL a times divisor := mul(22, 12)

Mobius inversion ORIGINAL inverse F(n/23) f(12) ORIGINAL a times divisor = 264

Mobius inversion ORIGINAL inverse F(n/23) f(12) plus ORIGINAL b := sum(264, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(12) plus ORIGINAL b = 265

Mobius inversion ORIGINAL inverse F(n/23) f(24) ORIGINAL a times divisor := mul(22, 24)

Mobius inversion ORIGINAL inverse F(n/23) f(24) ORIGINAL a times divisor = 528

Mobius inversion ORIGINAL inverse F(n/23) f(24) plus ORIGINAL b := sum(528, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(24) plus ORIGINAL b = 529

Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum := sum(23, 45, 67, 89, 133, 177, 265, 529)

Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum = 1328

Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) := mul(-1, 1328)

Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) = -1328

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor24:actual positive input for mu=24

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=24>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(24, 2)

upf extract divisor2 occurrence0 = 12

upf extract divisor2 occurrence1 := div(12, 2)

upf extract divisor2 occurrence1 = 6

upf extract divisor2 occurrence2 := div(6, 2)

upf extract divisor2 occurrence2 = 3

next divisor=2;previous remaining=24;all smaller divisors exhausted

extracted multiplicity=3;next remaining=3;remaining modulo2=1;extracted multiset=[2, 2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

upf canonical complete product factor3 := mul(8, 3)

upf canonical complete product factor3 = 24

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=24

complete increasing prime multiset=[2, 2, 2, 3];full product=24;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor24:input=24;COMPLETE prime multiset=[2, 2, 2, 3];distinct primes=[2, 3]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor24 := div(552, 24)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor24 = 23

Mobius inversion ORIGINAL inverse F(n/24):actual positive input=23;check complete integer range1..23

ALL positive divisors INCREASING=[1, 23];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/24) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/24) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/24) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/24) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/24) f(23) ORIGINAL a times divisor := mul(22, 23)

Mobius inversion ORIGINAL inverse F(n/24) f(23) ORIGINAL a times divisor = 506

Mobius inversion ORIGINAL inverse F(n/24) f(23) plus ORIGINAL b := sum(506, 1)

Mobius inversion ORIGINAL inverse F(n/24) f(23) plus ORIGINAL b = 507

Mobius inversion ORIGINAL inverse F(n/24) full forward divisor sum := sum(23, 507)

Mobius inversion ORIGINAL inverse F(n/24) full forward divisor sum = 530

Mobius inversion ORIGINAL inverse mu(24) times SAME F(n/24) := mul(0, 530)

Mobius inversion ORIGINAL inverse mu(24) times SAME F(n/24) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor46:actual positive input for mu=46

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=46>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(46, 2)

upf extract divisor2 occurrence0 = 23

next divisor=2;previous remaining=46;all smaller divisors exhausted

extracted multiplicity=1;next remaining=23;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo3=2;extracted multiset=[2]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 23)

upf canonical complete product factor1 = 46

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=46

complete increasing prime multiset=[2, 23];full product=46;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor46:input=46;COMPLETE prime multiset=[2, 23];distinct primes=[2, 23]

mu=1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor46 := div(552, 46)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor46 = 12

Mobius inversion ORIGINAL inverse F(n/46):actual positive input=12;check complete integer range1..12

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/46) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/46) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/46) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/46) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/46) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/46) f(3) ORIGINAL a times divisor := mul(22, 3)

Mobius inversion ORIGINAL inverse F(n/46) f(3) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/46) f(3) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(3) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/46) f(4) ORIGINAL a times divisor := mul(22, 4)

Mobius inversion ORIGINAL inverse F(n/46) f(4) ORIGINAL a times divisor = 88

Mobius inversion ORIGINAL inverse F(n/46) f(4) plus ORIGINAL b := sum(88, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(4) plus ORIGINAL b = 89

Mobius inversion ORIGINAL inverse F(n/46) f(6) ORIGINAL a times divisor := mul(22, 6)

Mobius inversion ORIGINAL inverse F(n/46) f(6) ORIGINAL a times divisor = 132

Mobius inversion ORIGINAL inverse F(n/46) f(6) plus ORIGINAL b := sum(132, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(6) plus ORIGINAL b = 133

Mobius inversion ORIGINAL inverse F(n/46) f(12) ORIGINAL a times divisor := mul(22, 12)

Mobius inversion ORIGINAL inverse F(n/46) f(12) ORIGINAL a times divisor = 264

Mobius inversion ORIGINAL inverse F(n/46) f(12) plus ORIGINAL b := sum(264, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(12) plus ORIGINAL b = 265

Mobius inversion ORIGINAL inverse F(n/46) full forward divisor sum := sum(23, 45, 67, 89, 133, 265)

Mobius inversion ORIGINAL inverse F(n/46) full forward divisor sum = 622

Mobius inversion ORIGINAL inverse mu(46) times SAME F(n/46) := mul(1, 622)

Mobius inversion ORIGINAL inverse mu(46) times SAME F(n/46) = 622

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor69:actual positive input for mu=69

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=69>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

next divisor=2;previous remaining=69;all smaller divisors exhausted

extracted multiplicity=0;next remaining=69;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(69, 3)

upf extract divisor3 occurrence0 = 23

next divisor=3;previous remaining=69;all smaller divisors exhausted

extracted multiplicity=1;next remaining=23;remaining modulo3=2;extracted multiset=[3]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[3]

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 23)

upf canonical complete product factor1 = 69

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=69

complete increasing prime multiset=[3, 23];full product=69;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor69:input=69;COMPLETE prime multiset=[3, 23];distinct primes=[3, 23]

mu=1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor69 := div(552, 69)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor69 = 8

Mobius inversion ORIGINAL inverse F(n/69):actual positive input=8;check complete integer range1..8

ALL positive divisors INCREASING=[1, 2, 4, 8];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/69) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/69) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/69) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/69) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/69) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/69) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/69) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/69) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/69) f(4) ORIGINAL a times divisor := mul(22, 4)

Mobius inversion ORIGINAL inverse F(n/69) f(4) ORIGINAL a times divisor = 88

Mobius inversion ORIGINAL inverse F(n/69) f(4) plus ORIGINAL b := sum(88, 1)

Mobius inversion ORIGINAL inverse F(n/69) f(4) plus ORIGINAL b = 89

Mobius inversion ORIGINAL inverse F(n/69) f(8) ORIGINAL a times divisor := mul(22, 8)

Mobius inversion ORIGINAL inverse F(n/69) f(8) ORIGINAL a times divisor = 176

Mobius inversion ORIGINAL inverse F(n/69) f(8) plus ORIGINAL b := sum(176, 1)

Mobius inversion ORIGINAL inverse F(n/69) f(8) plus ORIGINAL b = 177

Mobius inversion ORIGINAL inverse F(n/69) full forward divisor sum := sum(23, 45, 89, 177)

Mobius inversion ORIGINAL inverse F(n/69) full forward divisor sum = 334

Mobius inversion ORIGINAL inverse mu(69) times SAME F(n/69) := mul(1, 334)

Mobius inversion ORIGINAL inverse mu(69) times SAME F(n/69) = 334

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor92:actual positive input for mu=92

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=92>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(92, 2)

upf extract divisor2 occurrence0 = 46

upf extract divisor2 occurrence1 := div(46, 2)

upf extract divisor2 occurrence1 = 23

next divisor=2;previous remaining=92;all smaller divisors exhausted

extracted multiplicity=2;next remaining=23;remaining modulo2=1;extracted multiset=[2, 2]

next divisor=3;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo3=2;extracted multiset=[2, 2]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 23)

upf canonical complete product factor2 = 92

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=92

complete increasing prime multiset=[2, 2, 23];full product=92;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor92:input=92;COMPLETE prime multiset=[2, 2, 23];distinct primes=[2, 23]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor92 := div(552, 92)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor92 = 6

Mobius inversion ORIGINAL inverse F(n/92):actual positive input=6;check complete integer range1..6

ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/92) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/92) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/92) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/92) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/92) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/92) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/92) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/92) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/92) f(3) ORIGINAL a times divisor := mul(22, 3)

Mobius inversion ORIGINAL inverse F(n/92) f(3) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/92) f(3) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/92) f(3) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/92) f(6) ORIGINAL a times divisor := mul(22, 6)

Mobius inversion ORIGINAL inverse F(n/92) f(6) ORIGINAL a times divisor = 132

Mobius inversion ORIGINAL inverse F(n/92) f(6) plus ORIGINAL b := sum(132, 1)

Mobius inversion ORIGINAL inverse F(n/92) f(6) plus ORIGINAL b = 133

Mobius inversion ORIGINAL inverse F(n/92) full forward divisor sum := sum(23, 45, 67, 133)

Mobius inversion ORIGINAL inverse F(n/92) full forward divisor sum = 268

Mobius inversion ORIGINAL inverse mu(92) times SAME F(n/92) := mul(0, 268)

Mobius inversion ORIGINAL inverse mu(92) times SAME F(n/92) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor138:actual positive input for mu=138

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=138>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(138, 2)

upf extract divisor2 occurrence0 = 69

next divisor=2;previous remaining=138;all smaller divisors exhausted

extracted multiplicity=1;next remaining=69;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(69, 3)

upf extract divisor3 occurrence1 = 23

next divisor=3;previous remaining=69;all smaller divisors exhausted

extracted multiplicity=1;next remaining=23;remaining modulo3=2;extracted multiset=[2, 3]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[2, 3]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

upf canonical complete product factor2 := mul(6, 23)

upf canonical complete product factor2 = 138

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=138

complete increasing prime multiset=[2, 3, 23];full product=138;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor138:input=138;COMPLETE prime multiset=[2, 3, 23];distinct primes=[2, 3, 23]

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor138 := div(552, 138)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor138 = 4

Mobius inversion ORIGINAL inverse F(n/138):actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/138) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/138) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/138) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/138) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/138) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/138) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/138) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/138) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/138) f(4) ORIGINAL a times divisor := mul(22, 4)

Mobius inversion ORIGINAL inverse F(n/138) f(4) ORIGINAL a times divisor = 88

Mobius inversion ORIGINAL inverse F(n/138) f(4) plus ORIGINAL b := sum(88, 1)

Mobius inversion ORIGINAL inverse F(n/138) f(4) plus ORIGINAL b = 89

Mobius inversion ORIGINAL inverse F(n/138) full forward divisor sum := sum(23, 45, 89)

Mobius inversion ORIGINAL inverse F(n/138) full forward divisor sum = 157

Mobius inversion ORIGINAL inverse mu(138) times SAME F(n/138) := mul(-1, 157)

Mobius inversion ORIGINAL inverse mu(138) times SAME F(n/138) = -157

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor184:actual positive input for mu=184

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=184>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(184, 2)

upf extract divisor2 occurrence0 = 92

upf extract divisor2 occurrence1 := div(92, 2)

upf extract divisor2 occurrence1 = 46

upf extract divisor2 occurrence2 := div(46, 2)

upf extract divisor2 occurrence2 = 23

next divisor=2;previous remaining=184;all smaller divisors exhausted

extracted multiplicity=3;next remaining=23;remaining modulo2=1;extracted multiset=[2, 2, 2]

next divisor=3;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo3=2;extracted multiset=[2, 2, 2]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[2, 2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

upf canonical complete product factor3 := mul(8, 23)

upf canonical complete product factor3 = 184

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=184

complete increasing prime multiset=[2, 2, 2, 23];full product=184;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor184:input=184;COMPLETE prime multiset=[2, 2, 2, 23];distinct primes=[2, 23]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor184 := div(552, 184)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor184 = 3

Mobius inversion ORIGINAL inverse F(n/184):actual positive input=3;check complete integer range1..3

ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/184) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/184) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/184) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/184) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/184) f(3) ORIGINAL a times divisor := mul(22, 3)

Mobius inversion ORIGINAL inverse F(n/184) f(3) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/184) f(3) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/184) f(3) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/184) full forward divisor sum := sum(23, 67)

Mobius inversion ORIGINAL inverse F(n/184) full forward divisor sum = 90

Mobius inversion ORIGINAL inverse mu(184) times SAME F(n/184) := mul(0, 90)

Mobius inversion ORIGINAL inverse mu(184) times SAME F(n/184) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor276:actual positive input for mu=276

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=276>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(276, 2)

upf extract divisor2 occurrence0 = 138

upf extract divisor2 occurrence1 := div(138, 2)

upf extract divisor2 occurrence1 = 69

next divisor=2;previous remaining=276;all smaller divisors exhausted

extracted multiplicity=2;next remaining=69;remaining modulo2=1;extracted multiset=[2, 2]

upf extract divisor3 occurrence2 := div(69, 3)

upf extract divisor3 occurrence2 = 23

next divisor=3;previous remaining=69;all smaller divisors exhausted

extracted multiplicity=1;next remaining=23;remaining modulo3=2;extracted multiset=[2, 2, 3]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[2, 2, 3]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 3)

upf canonical complete product factor2 = 12

upf canonical complete product factor3 := mul(12, 23)

upf canonical complete product factor3 = 276

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=276

complete increasing prime multiset=[2, 2, 3, 23];full product=276;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor276:input=276;COMPLETE prime multiset=[2, 2, 3, 23];distinct primes=[2, 3, 23]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor276 := div(552, 276)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor276 = 2

Mobius inversion ORIGINAL inverse F(n/276):actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/276) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/276) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/276) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/276) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/276) f(2) ORIGINAL a times divisor := mul(22, 2)

Mobius inversion ORIGINAL inverse F(n/276) f(2) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/276) f(2) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/276) f(2) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/276) full forward divisor sum := sum(23, 45)

Mobius inversion ORIGINAL inverse F(n/276) full forward divisor sum = 68

Mobius inversion ORIGINAL inverse mu(276) times SAME F(n/276) := mul(0, 68)

Mobius inversion ORIGINAL inverse mu(276) times SAME F(n/276) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor552:actual positive input for mu=552

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=552>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(552, 2)

upf extract divisor2 occurrence0 = 276

upf extract divisor2 occurrence1 := div(276, 2)

upf extract divisor2 occurrence1 = 138

upf extract divisor2 occurrence2 := div(138, 2)

upf extract divisor2 occurrence2 = 69

next divisor=2;previous remaining=552;all smaller divisors exhausted

extracted multiplicity=3;next remaining=69;remaining modulo2=1;extracted multiset=[2, 2, 2]

upf extract divisor3 occurrence3 := div(69, 3)

upf extract divisor3 occurrence3 = 23

next divisor=3;previous remaining=69;all smaller divisors exhausted

extracted multiplicity=1;next remaining=23;remaining modulo3=2;extracted multiset=[2, 2, 2, 3]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[2, 2, 2, 3]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

upf canonical complete product factor3 := mul(8, 3)

upf canonical complete product factor3 = 24

upf canonical complete product factor4 := mul(24, 23)

upf canonical complete product factor4 = 552

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=552

complete increasing prime multiset=[2, 2, 2, 3, 23];full product=552;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor552:input=552;COMPLETE prime multiset=[2, 2, 2, 3, 23];distinct primes=[2, 3, 23]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor552 := div(552, 552)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor552 = 1

Mobius inversion ORIGINAL inverse F(n/552):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/552) f(1) ORIGINAL a times divisor := mul(22, 1)

Mobius inversion ORIGINAL inverse F(n/552) f(1) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/552) f(1) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/552) f(1) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/552) full forward divisor sum := sum(23)

Mobius inversion ORIGINAL inverse F(n/552) full forward divisor sum = 23

Mobius inversion ORIGINAL inverse mu(552) times SAME F(n/552) := mul(0, 23)

Mobius inversion ORIGINAL inverse mu(552) times SAME F(n/552) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(31696, -14796, -7928, 0, 3702, 0, 0, -1328, 0, 622, 334, 0, -157, 0, 0, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 12145

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(22, 552)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 12144

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(12144, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 12145

Mobius inversion ORIGINAL inverse:actual n=552;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 552, 31696, 31696], [2, -1, 276, 14796, -14796], [3, -1, 184, 7928, -7928], [4, 0, 138, 6344, 0], [6, 1, 92, 3702, 3702], [8, 0, 69, 2116, 0], [12, 0, 46, 1588, 0], [23, -1, 24, 1328, -1328], [24, 0, 23, 530, 0], [46, 1, 12, 622, 622], [69, 1, 8, 334, 334], [92, 0, 6, 268, 0], [138, -1, 4, 157, -157], [184, 0, 3, 90, 0], [276, 0, 2, 68, 0], [552, 0, 1, 23, 0]]

complete reconstructed=12145;direct f(n)=12145;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-14796, -7928, 0, 3702, 0, 0, -1328, 0, 622, 334, 0, -157, 0, 0, 0)

Mobius inversion ORIGINAL omitted-divisor partial sum = -19551

Mobius inversion complete minus partial := sub(12145, -19551)

Mobius inversion complete minus partial = 31696

ORIGINAL full divisors=[1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552]

complete=12145=f(n)=12145;partial=-19551;lost term=31696!=0;incomplete_certificate fixed No

逐步反向核對

• S411 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S410 · premise_and_result_audit — 以精確有理數重算 Mobius inversion complete minus partial，不以小數近似替代。
• S409 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL omitted-divisor partial sum，不以小數近似替代。
• S408 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S407 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct plus ORIGINAL b，不以小數近似替代。
• S406 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct ORIGINAL a times n，不以小數近似替代。
• S405 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse complete inverse-term sum，不以小數近似替代。
• S404 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(552) times SAME F(n/552)，不以小數近似替代。
• S403 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/552) full forward divisor sum，不以小數近似替代。
• S402 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/552) f(1) plus ORIGINAL b，不以小數近似替代。
• S401 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/552) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S400 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S399 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor552，不以小數近似替代。
• S398 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S397 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S396 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor4，不以小數近似替代。
• S395 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S394 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S393 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S392 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S391 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S390 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S389 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence3，不以小數近似替代。
• S388 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S387 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S386 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S385 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S384 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S383 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S382 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(276) times SAME F(n/276)，不以小數近似替代。
• S381 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/276) full forward divisor sum，不以小數近似替代。
• S380 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/276) f(2) plus ORIGINAL b，不以小數近似替代。
• S379 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/276) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S378 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/276) f(1) plus ORIGINAL b，不以小數近似替代。
• S377 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/276) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S376 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S375 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor276，不以小數近似替代。
• S374 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S373 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S372 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S371 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S370 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S369 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S368 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S367 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S366 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence2，不以小數近似替代。
• S365 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S364 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S363 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S362 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S361 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S360 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(184) times SAME F(n/184)，不以小數近似替代。
• S359 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/184) full forward divisor sum，不以小數近似替代。
• S358 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/184) f(3) plus ORIGINAL b，不以小數近似替代。
• S357 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/184) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S356 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/184) f(1) plus ORIGINAL b，不以小數近似替代。
• S355 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/184) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S354 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S353 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor184，不以小數近似替代。
• S352 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S351 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S350 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S349 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S348 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S347 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S346 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S345 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S344 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S343 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S342 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S341 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S340 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S339 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S338 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(138) times SAME F(n/138)，不以小數近似替代。
• S337 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/138) full forward divisor sum，不以小數近似替代。
• S336 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/138) f(4) plus ORIGINAL b，不以小數近似替代。
• S335 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/138) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S334 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/138) f(2) plus ORIGINAL b，不以小數近似替代。
• S333 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/138) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S332 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/138) f(1) plus ORIGINAL b，不以小數近似替代。
• S331 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/138) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S330 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S329 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor138，不以小數近似替代。
• S328 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S327 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S326 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S325 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S324 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S323 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S322 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S321 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S320 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S319 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S318 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S317 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S316 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(92) times SAME F(n/92)，不以小數近似替代。
• S315 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) full forward divisor sum，不以小數近似替代。
• S314 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) f(6) plus ORIGINAL b，不以小數近似替代。
• S313 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S312 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) f(3) plus ORIGINAL b，不以小數近似替代。
• S311 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S310 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) f(2) plus ORIGINAL b，不以小數近似替代。
• S309 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S308 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) f(1) plus ORIGINAL b，不以小數近似替代。
• S307 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/92) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S306 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S305 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor92，不以小數近似替代。
• S304 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S303 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S302 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S301 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S300 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S299 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S298 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S297 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S296 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S295 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S294 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S293 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S292 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(69) times SAME F(n/69)，不以小數近似替代。
• S291 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) full forward divisor sum，不以小數近似替代。
• S290 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) f(8) plus ORIGINAL b，不以小數近似替代。
• S289 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) f(8) ORIGINAL a times divisor，不以小數近似替代。
• S288 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) f(4) plus ORIGINAL b，不以小數近似替代。
• S287 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S286 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) f(2) plus ORIGINAL b，不以小數近似替代。
• S285 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S284 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) f(1) plus ORIGINAL b，不以小數近似替代。
• S283 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/69) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S282 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S281 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor69，不以小數近似替代。
• S280 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S279 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S278 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S277 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S276 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S275 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S274 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。
• S273 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S272 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S271 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S270 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(46) times SAME F(n/46)，不以小數近似替代。
• S269 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) full forward divisor sum，不以小數近似替代。
• S268 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(12) plus ORIGINAL b，不以小數近似替代。
• S267 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(12) ORIGINAL a times divisor，不以小數近似替代。
• S266 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(6) plus ORIGINAL b，不以小數近似替代。
• S265 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S264 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(4) plus ORIGINAL b，不以小數近似替代。
• S263 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S262 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(3) plus ORIGINAL b，不以小數近似替代。
• S261 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S260 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(2) plus ORIGINAL b，不以小數近似替代。
• S259 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S258 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(1) plus ORIGINAL b，不以小數近似替代。
• S257 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S256 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S255 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor46，不以小數近似替代。
• S254 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S253 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S252 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S251 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S250 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S249 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S248 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S247 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S246 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S245 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S244 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(24) times SAME F(n/24)，不以小數近似替代。
• S243 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/24) full forward divisor sum，不以小數近似替代。
• S242 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/24) f(23) plus ORIGINAL b，不以小數近似替代。
• S241 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/24) f(23) ORIGINAL a times divisor，不以小數近似替代。
• S240 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/24) f(1) plus ORIGINAL b，不以小數近似替代。
• S239 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/24) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S238 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S237 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor24，不以小數近似替代。
• S236 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S235 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S234 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S233 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S232 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S231 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S230 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S229 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S228 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S227 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S226 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S225 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S224 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23)，不以小數近似替代。
• S223 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum，不以小數近似替代。
• S222 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(24) plus ORIGINAL b，不以小數近似替代。
• S221 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(24) ORIGINAL a times divisor，不以小數近似替代。
• S220 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(12) plus ORIGINAL b，不以小數近似替代。
• S219 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(12) ORIGINAL a times divisor，不以小數近似替代。
• S218 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(8) plus ORIGINAL b，不以小數近似替代。
• S217 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(8) ORIGINAL a times divisor，不以小數近似替代。
• S216 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(6) plus ORIGINAL b，不以小數近似替代。
• S215 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S214 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(4) plus ORIGINAL b，不以小數近似替代。
• S213 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S212 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(3) plus ORIGINAL b，不以小數近似替代。
• S211 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S210 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(2) plus ORIGINAL b，不以小數近似替代。
• S209 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S208 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b，不以小數近似替代。
• S207 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S206 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S205 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23，不以小數近似替代。
• S204 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S203 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S202 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S201 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S200 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S199 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S198 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S197 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S196 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12)，不以小數近似替代。
• S195 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum，不以小數近似替代。
• S194 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(46) plus ORIGINAL b，不以小數近似替代。
• S193 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(46) ORIGINAL a times divisor，不以小數近似替代。
• S192 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(23) plus ORIGINAL b，不以小數近似替代。
• S191 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(23) ORIGINAL a times divisor，不以小數近似替代。
• S190 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(2) plus ORIGINAL b，不以小數近似替代。
• S189 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S188 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b，不以小數近似替代。
• S187 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S186 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S185 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12，不以小數近似替代。
• S184 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S183 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S182 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S181 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S180 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S179 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S178 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S177 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S176 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S175 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S174 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8)，不以小數近似替代。
• S173 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum，不以小數近似替代。
• S172 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(69) plus ORIGINAL b，不以小數近似替代。
• S171 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(69) ORIGINAL a times divisor，不以小數近似替代。
• S170 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(23) plus ORIGINAL b，不以小數近似替代。
• S169 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(23) ORIGINAL a times divisor，不以小數近似替代。
• S168 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(3) plus ORIGINAL b，不以小數近似替代。
• S167 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S166 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b，不以小數近似替代。
• S165 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S164 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S163 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8，不以小數近似替代。
• S162 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S161 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S160 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S159 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S158 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S157 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S156 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S155 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S153 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S152 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S151 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6)，不以小數近似替代。
• S150 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum，不以小數近似替代。
• S149 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(92) plus ORIGINAL b，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(92) ORIGINAL a times divisor，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(46) plus ORIGINAL b，不以小數近似替代。
• S146 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(46) ORIGINAL a times divisor，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(23) plus ORIGINAL b，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(23) ORIGINAL a times divisor，不以小數近似替代。
• S143 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(4) plus ORIGINAL b，不以小數近似替代。
• S142 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S137 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S136 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6，不以小數近似替代。
• S135 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S134 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S133 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S132 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S131 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S130 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S129 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S128 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S127 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4)，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(138) plus ORIGINAL b，不以小數近似替代。
• S124 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(138) ORIGINAL a times divisor，不以小數近似替代。
• S123 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(69) plus ORIGINAL b，不以小數近似替代。
• S122 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(69) ORIGINAL a times divisor，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(46) plus ORIGINAL b，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(46) ORIGINAL a times divisor，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(23) plus ORIGINAL b，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(23) ORIGINAL a times divisor，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(6) plus ORIGINAL b，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S109 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S108 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4，不以小數近似替代。
• S107 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S106 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S105 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S103 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S102 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S101 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S100 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S099 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S098 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3)，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(184) plus ORIGINAL b，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(184) ORIGINAL a times divisor，不以小數近似替代。
• S094 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(92) plus ORIGINAL b，不以小數近似替代。
• S093 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(92) ORIGINAL a times divisor，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(46) plus ORIGINAL b，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(46) ORIGINAL a times divisor，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(23) plus ORIGINAL b，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(23) ORIGINAL a times divisor，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(8) plus ORIGINAL b，不以小數近似替代。
• S087 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(8) ORIGINAL a times divisor，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S080 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S079 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3，不以小數近似替代。
• S078 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S077 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S076 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S075 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S074 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S073 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(276) plus ORIGINAL b，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(276) ORIGINAL a times divisor，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(138) plus ORIGINAL b，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(138) ORIGINAL a times divisor，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(92) plus ORIGINAL b，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(92) ORIGINAL a times divisor，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(69) plus ORIGINAL b，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(69) ORIGINAL a times divisor，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(46) plus ORIGINAL b，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(46) ORIGINAL a times divisor，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(23) plus ORIGINAL b，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(23) ORIGINAL a times divisor，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(12) plus ORIGINAL b，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(12) ORIGINAL a times divisor，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S047 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S046 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S045 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S044 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S043 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S040 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(552) plus ORIGINAL b，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(552) ORIGINAL a times divisor，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(276) plus ORIGINAL b，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(276) ORIGINAL a times divisor，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(184) plus ORIGINAL b，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(184) ORIGINAL a times divisor，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(138) plus ORIGINAL b，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(138) ORIGINAL a times divisor，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(92) plus ORIGINAL b，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(92) ORIGINAL a times divisor，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(69) plus ORIGINAL b，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(69) ORIGINAL a times divisor，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(46) plus ORIGINAL b，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(46) ORIGINAL a times divisor，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(24) plus ORIGINAL b，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(24) ORIGINAL a times divisor，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(23) ORIGINAL a times divisor，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(8) plus ORIGINAL b，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(8) ORIGINAL a times divisor，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(552, [1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 1, [2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 12145, -19551, 12145, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。